Linear system analysis has long been an indispensable tool for analog circuit designers. In order to benefit from such analysis, circuits are typically analyzed in their small-signal models from which various linear system parameters, such as poles, zeros, gain, and bandwidth, can be derived. Linear analyses, such as AC and periodic AC (PAC) analyses, effectively simulate the linearized response of the circuits. Due to the prevalence of linear analysis tools, analog circuit designers often strive to achieve certain linear system behaviors while minimizing nonlinearities such as offset and distortion. Linear analysis, however, is not easily applied to mixed-signal circuits because mixed-signal circuits do not have a steady-state response that is either time-invariant or periodically time-varying (PTV).
It is a steady trend in integrated circuit (IC) design to implement increasingly more analog circuits into mixed-signal systems with the digital circuits either replacing or assisting the analog circuits, due in part to CMOS transistors being poor gain elements but good, fast switches. For example, by leveraging the fast speed of digital logic, delta-sigma (ΔΣ) data converters can achieve high resolutions without requiring the stringent accuracy required for analog circuits, namely, the uniform distribution of the effective threshold levels in data conversions between analog and digital domains. Also, many phase-locked loops (PLLs) now implement time-to-digital converters and digital-controlled oscillators so that the loop filters can be realized entirely in digital logic, thus reducing the large filter area and enabling more sophisticated control of the loop dynamics. Furthermore, many analog front-end circuits in wireless and wired communication systems are equipped with digital calibration loops that monitor and correct undesired properties of the circuits such as offset, skew, or duty-cycle error.
A common characteristic of these emerging mixed-signal systems is random behavior. For example, most digitally-controlled feedback loops including digital PLLs, delayed-locked loops (DLLs), and calibration loops show a periodic dithering when approaching the steady states rather than the exponential convergence shown by linear circuits, meaning the feedback systems do not settle into a constant state but rather alternate among multiple discrete states. Sometimes this randomness is even intentional. For example, ΔΣ modulators rely on randomness to shape the deterministic quantization errors to out-of-band random noise. Furthermore, injecting pseudo-random dithering into ΔΣ data converters has been found to improve linearity and suppress periodic idle tones. Dynamic element matching is another technique that randomizes the selection of the sub-elements so that any mismatch among them appears as random noise. Also in digital calibration loops, it has been reported that randomizing the calibration period can help reduce undesired harmonic tones that arise due to periodicity.
Despite the fact that many mixed-signal systems are stochastic, their design intents are still linear and simulating their adjoint linear system responses at the steady-states (e.g. the frequency-domain transfer function) is of high interest to the designers. An adjoint linear system refers to a hypothetical linear system which describes the responses of the original system at the designated steady state (also called operating point) for infinitesimal changes in an input signals. For example, the ΔΣ data converters are designed with specific signal transfer functions (STF) and noise transfer functions (NTF) to maximize the signal-to-noise ratio (SNR). Also, phase transfer functions are the most common way of describing the characteristics, such as bandwidth and jitter peaking, of a digital PLL. While AC and PAC analyses available in today's circuit simulators can efficiently simulate the adjoint linear responses for the circuits that have DC or periodic steady-states, these simulators cannot be applied to the above-mentioned mixed-signal systems that have stochastic steady-states.
Due to the unavailability of AC and PAC analyses, circuit designers often use time-domain analysis, also referred to as transient analysis, to simulate mixed-signal systems. The time required to perform transient analysis on complicated circuit designs, however, is often thousands or even hundreds of thousands times greater than the time required to perform AC or PAC analyses, and even with this increased testing time, the simulating achieved by transient analysis is less complete than that of AC or PAC analyses. It is therefore desirable to extend steady-state and AC analyses to mixed-signal systems with Stochastic steady states.